On Gromov-Witten theory of root gerbes

TL;DR

This work advances the understanding of Gromov-Witten theory for root gerbes by giving a complete genus $0$ computation for $\mu_r$-root gerbes over smooth bases and establishing a precise decomposition into base theory under a change of variables. It constructs the moduli of twisted stable maps to root gerbes, derives a pushforward formula linking the virtual fundamental classes to those of the base, and proves a genus $0 decomposition that expresses $\mathcal{F}^0_{\mathcal{G}}$ in terms of $\mathcal{F}^0_X$ with a representation-parameterized twist $Q_{\rho}$. For toric gerbes, the paper verifies the decomposition conjecture in all genera by reducing toric GW theory to that of the reduced base via explicit variable changes on cohomology and Novikov variables, and discusses extensions to general abelian gerbes and concrete examples such as weighted projective lines. The results provide a rigorous bridge between Gromov-Witten theory of gerbes and their bases, with potential implications for computations in orbifold GW theory and toric geometry.

Abstract

This research announcement discusses our results on Gromov-Witten theory of root gerbes. A complete calculation of genus 0 Gromov-Witten theory of $μ_{r}$-root gerbes over a smooth base scheme is obtained by a direct analysis of virtual fundamental classes. Our result verifies the genus 0 part of the so-called decomposition conjecture which compares Gromov-Witten theory of étale gerbes with that of the bases. We also verify this conjecture in all genera for toric gerbes over toric Deligne-Mumford stacks.

Theorems & Definitions (9)

• Theorem 2.1
• Theorem 2.2
• Theorem 2.3
• Theorem 2.4
• Remark 2.5
• Theorem 3.1
• Theorem 3.2
• Remark 3.3
• Example 3.4